The marginal distribution of temperature extremes is changing -

Is the dependence also changing?

If so by how much?

What does this mean for events like heatwaves?

• Marginals
  

• Dependence
  

• Observations
  

• Climate Models

Let \(M\) hottest 3-day average temperature in a year: \[ \mathbb{P}(M < z) = \exp \left\{ - \left[ 1 + \xi \left(\dfrac{z-\mu}{\sigma}\right) \right]_+^{-1 / \xi} \right\},\] where \([v]_+ = \max \left\lbrace 0,v \right\rbrace\), \(\mu \in \mathbb{R}\), \(\sigma \in \mathbb{R}^+\) and \(\xi \in \mathbb{R}\).
(Fisher and Tippett 1928, Gnendenko 1943)

• Address the trend in changing temperature by fitting temporal covariates to our parameters

• Transform the GEV marginals to common GEV(1,1,1) marignal distributions:
  \(\mathbb{P}(M < z) = \exp(-1/z)\)

Now dependence!

Assume the distribution of \((M_{i}, M_{j})\) can be approximated with a bivariate extreme value theory distribution, then

\[\mathbb{P}\left( M_{i} \leq z_i,\, M_{j} \leq z_j \right) = \exp\left\{ - V_{ij} \left( \dfrac{-1}{\log F_i(z_i)}, \, \dfrac{-1}{\log F_j(z_j)}\right) \right\}\] where the exponent measure \(V_{ij}(a,b)\) is given by \[V_{ij}(a, b) = 2 \int_0^1 \max \left(\dfrac{w}{a}, \dfrac{1-w}{b} \right) \text{d}H_{ij}(w)\] and \(H\) is any distribution function on \([0,1]\) with expectation equal to 0.5.

Lots of parameteric options to specify form of \(V_{ij}(a, b)\), but

• Want to avoid assuming that a parametric form that is constant in time

• Want a solution that will work for lots of different pairs

so …

Nonparametric

\[V_{ij}(z_i, z_j) = \left(\frac{1}{z_i} + \frac{1}{z_j}\right)A\left(\frac{z_i}{|z_i + z_j|} + \frac{z_j}{|z_i + z_j|}\right)\]

C1: \(A(t)\) is continuous and convex for \(t \in [0,1]\)
C2: Bounds on the function of \(\max \{(1 - t), t \} \leq A(t) \leq 1\)
C3: Boundary conditions of \(A(0) = A(1) = 1\)

Model background:

• 22 Global Climate Model and Regional Climate Model combinations from EURO-CORDEX

• 0.11 degree resolation (upscale)

• rotated-pole grid

• 1950-2005 with historical forcing

• 2006-2100 with RCP 8.5

• Consider Annual Maxima

• Early: 1951 - 2000
  Middle: 2001 - 2050
  Late: 2051 - 2100
  Assume stationarity within each window (big assumption)

• Fit pairwise distributions relative to a central location

\[\mathbb{P}\left(M_{i} \leq z, M_{j} \leq z\right) = \left[\mathbb{P}(M_{i} \leq z)\mathbb{P}(M_{j} \leq z) \right]^{A\left(1/2\right)}.\]

Early: 1951 - 2000

Strong initial evidence to suggest:

• The dependence between temperature extremes is increasing

• Area impacted by heat extremes is getting larger

• Fit a time-varying Pickands dependence function

• Examine assumptions on the marginal distriubtion

• Examine whether models are accurately representing heat extremes

• Think about how to translate insights from extreme value statistics into insights about changing heat extremes